A sample space consists of 9 elementary event E1, E2, E3, ……E8, E9 whose probabilities are P(E1) = P(E2) = 0.08, P(E3) = P(E4) = 0.1, P(E6) = P(E7) = 0.2, P(E8) = P(E9) = 0.07

Suppose A = {E1, E5, E8}, B = {E2, E8, E9}


(i) Compute P(A), P(B) and P(A ∩ B).


(ii) Using the addition law of probability, find P (A ∪ B).


(iii) List the composition of the event A ∪ B, and calcite P(A ∪ B) by adding the probabilities of the elementary events.


(iv) Calculate from P(B), also calculate directly from the elementary events of .

Clearly according to questions sample space contains 9 elementary events(events with single outcome)


Let S represents the sample space.


∴ S = E1∪ E2∪ E3 ∪ ……∪ E8∪ E9


Given A = {E1, E5, E8}


Or A = E1∪ E5∪ E8


P(E1) = P(E2) = 0.08, P(E3) = P(E4) = 0.1, P(E6) = P(E7) = 0.2, P(E8) = P(E9) = 0.07


∴ P(A) = P(E1∪ E5∪ E8) = P(E1)+P(E5) + P(E8)


⇒ P(A) = 0.08 + P(E5) + 0.07 = 0.15 + P(E5) …(1)


P(E5) is missing, so we need to find it.


Given,


B = {E2, E8, E9} or B = E2∪ E8∪ E9


∴ P(B) = P(E2∪ E8∪ E9) = P(E2)+P(E8) + P(E9) = = 0.08+0.07+0.07 = 0.21


∴ P(B) = 0.21 ….ans(i)


∴ B’ = {E1, E3, E4, E5, E6, E7} or B’ = E1∪ E3∪ E4∪ E5∪ E6∪ E7


∴ P(B’) = P(E1) + P(E3) + P(E4) + P(E5) + P(E6) + P(E7)


1 – 0.21 = 0.08 + 0.1 + 0.1 + P(E5) + 0.2 + 0.2


⇒ 0.79 = 0.68 + P(E5)


∴ P(E5) = 0.79 – 0.68 = 0.11


∴ from equation 1, we get-


P(A) = 0.15 + P(E5) = 0.15 + 0.11 = 0.26 (i)


Clearly A ∩ B = {E8}


∴ P(A ∩ B) = P(E8) = 0.07 (i)


Using addition law of probability we know that-


P(A ∪ B) = P(A) + P(B) – P(A ∩ B)


⇒ P(A ∪ B) = 0.26 + 0.21 – 0.07


∴ P(A ∪ B) = 0.4 (ii)


As, A ∪ B = {E1, E5, E8} ∪ {E2, E8, E9}


⇒ A ∪ B = {E1, E5, E8, E2, E9}


∴ P(A ∪ B) = P(E1)+ P(E5)+ P(E8)+ P(E2)+ P(E9)


⇒ P(A ∪ B) = 0.08 + 0.11 + 0.07 + 0.08 + 0.07 = 0.41


∴ P(A ∪ B) = 0.41 (iii)


As , P(B) = 0.21


∴ P(B’) = 1 – 0.21 = 0.79 (iv)


Calculation of P(B’) using sets –


B’ = {E1, E3, E4, E5, E6, E7} or B’ = E1∪ E3∪ E4∪ E5∪ E6∪ E7


∴ P(B’) = P(E1) + P(E3) + P(E4) + P(E5) + P(E6) + P(E7)


P(B’) = 0.08 + 0.1 + 0.1 + 0.11 + 0.2 + 0.2


= 0.79 (iv)


Clearly through both the ways we get the same answer.


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